1. Lecture 16:
Hilbert-Huang Transform
Background:
An examination of Fourier Analysis
Existing non-stationary data handling method
Instantaneous frequency
Intrinsic mode functions(IMF)
Empirical mode decomposition(EMD)
Mathematical considerations

2. HHT Transform Sources:
Original Paper:
Huang, et al. "The empirical mode decomposition and the Hilbert spectrum for
nonlinear and non-stationary time series analysis." Proc. R. Soc. Lond. A (1998) 454, 903–995
NASA:
Book: HILBERT-HUANG TRANSFORM AND ITS APPLICATIONS
Ed. by Norden E Huang and Samuel S P Shen
EMD code:
Other goodies:
HHT-based Identification codes:

3. Why not the Fourier analysis (FA)?
The FA performs well when the system is linear;
Measure at least two output: y1(t) and y2(t) corresponding
to input x1(t) and x2(t).
Now apply input: x(t) = a x1(t) + b x2(t) + c x3(t) +…
if the output is given by y(t) = a y1(t) + b y2(t) + c y3(t) + …
then the system is deemed to be linear,
And when data are periodic or stationary;

4. And when is the FA not so best?
when data are nonstationary;
the FA basis functions are global, hence they cannot treat
local nonlinearity without significant dispersions (spreading);
The above is especially true when the wave forms deviate
significantly from sinusoidal form;
For delta function-like waves, an excessive number of
harmonic terms are required, let alone the Gibbs phenomena.

5. Nonstationary data processing methods
Spectrogram
Wavelets analysis
Wigner-Ville distribution
Evolutionary spectrum
Empirical orthogonal function expansion (EOF)
Smoothed moving average
Trend least-squares estimation

6. Instantaneous Frequency
Definition of Hilbert Transform:
Complexification:
Frequency:

7. Instantaneous frequency - cont’d
The instantaneous frequency defined is a scalar; which
means that w is a monocomponent. In reality, the signal
may not represent a monocomponent. Therefore, one should
interpret it as a localized frequency within a narrow band.
As the concept of bandwidth plays a crucial role, we borrow
its definition from the signal processing:
The number of zero crossing per unit time is given by
while the expected number of extrema per unit time is
given by
where mi is the i-th moment of the spectrum.

8. Instantaneous frequency - cont’d
Hence, a standard bandwidth measure can be given by
n2 = p2 ( N21 - N20 )
Note that if n =0, the expected numbers of extrema and zero
crossings are equal. It is this observation we will exploit in
the empirical mode decomposition later on.
However, the instantaneous frequency defined previously still
yields a global measure. Hence, when one decomposes the signal
into multi-components, a key criterion is to ensure the associated
frequency is locally valid. This is discussed in the next,
Intrinsic Mode Functions.

9. Intrinsic Mode Functions (IMF)
(implies oscillations embedded in the data)
Suppose a function is symmetric with respect to the local zero
mean, and have the same numbers of extrema and zero crossing.
Then a physically meaningful local instantaneous frequency can
be discerned from the function.
Exploiting this concept, an intrinsic mode function satisfies the
following two conditions:
In the whole data set, the number of extrema and the number
of zero crossings must either be equal or differ at most by one;
(adaptation of narrow band concept)
At any point, the mean value of the envelope defined by the
local maxima and the envelope defined by the local minima
is zero (new - adoption of local properties).

10. Intrinsic Mode Functions - Cont’d
Modification of local mean: the local mean of the two envelopes
defined by the local maxima and local minima - this forces
the local symmetry. However, it does engender an alias in the
instantaneous frequency for nonlinearly deformed waves.
The IMF properties:
each IMF involves only one mode of oscillation;
each IMF characterizes not only a narrow band but
both amplitude and frequency modulations;
an IMF can thus be nonstationary.

11. Huang et al Statement on why IMF-based instantaneous frequency
makes sense (Proc. R. Soc. Lond. A (1998), p.916):

12. Empirical mode decomposition method (EMD) in a nutshell
EMD identifies the intrinsic oscillatory modes by their
characteristic time scales in the data empirically, then decomposes
the data into the corresponding IMFs via the sifting process.
Thus, it is an algorithm to assign an instantaneous frequency
to each IMF in order to decompose an arbitrary set of data;
this means, for complex data, we can allow more than one
instantaneous frequency at a time locally. In doing so we obtain
IMFs as most data do not consist of IMFs.
In other words, EMD decomposes an arbitrary data set, whether
they are linear, nonlinear or nonstationary, into a set of IMFs.

13. Assumptions introduced in EMD
The signal has at least two extrema - one maxima
and one minima;
The characteristic time scale is defined by the time lapse
between the extrema;
If the data is totally devoid of extrema but contained only
inflection points, then it can be differentiated once or more
times to reveal the extrema. Final results then can be obtained
by integration(s) of the components.

14. The sifting process

16. Signal from Final Shifting for c1

17. Algorithmic statement of the visual process demonstrated so far
1. For data X(t), we mark the local maxima and local minima, and
interpolate the extrema points via, e.g., splines to obtain upper
And lower envelopes.
2. Obtain the mean of the two envelopes, m1 .
3. Obtain h1 = X(t) - m1 and inspect whether the the number of
extrema and the number of zero crossings must either be equal
or differ at most by one. Plus, inspect whether all the local maxima
are positive and all the local minima are negative.
3. If not, repeat the sifting process and obtain h1 - m11 = h11.
and repeat to obtain
h1(k-1) - m1k = h1k

18. Algorithmic statement - cont’d
If h1k constitutes an IMF, then designate it c1 = h1k.
Now we obtain the first residual r1 via
r1 = X(t) - c1
Treat r1 as a new data set, and perform the sifting process
to obtain c2.
Continuing the sifting process we obtain
r2 = r1 - c2, …, rn-1 - cn = rn.
Finally, the original signal is decomposed in terms of IMFs:

19. Hilbert-Huang Transform (HHT) for Nonlinearity Detection

20. HHT - Cont’d
The nonlinear and non-stationary time series signal reconstructed via the HHT are:
complete by employing its stopping criterion;
nearly orthogonal;
local; and,
highly adaptive.
We label these properties into one phrase COLA.
We will now introduce the HHT via the example problem used
by Huang and et al.

21. 1. The signal has at least one maximum and one minimum;
HHT - Cont’d
The Sifting Process
Assumptions introduced (Huang et al, 1998):
1. The signal has at least one maximum and one minimum;
2. The characteristic time scale is defined by the time lapse
between the extrema;
3. If the data were totally devoid of extrema but contained only
inflection points, then it can be differentiated once or more
times to reveal extrema; and, final results can be obtained
by integration(s) of the components.

22. In addition to the above assumptions, they imposed a restriction
The Sifting Process
Restriction imposed (Huang et al, 1998):
HHT - Cont’d
In addition to the above assumptions, they imposed a restriction
that the the resulting intrinsic mode functions (IMF) be
symmetric locally with respect to the zero mean level.
This restriction implies the IMFs have the same numbers of
zero-crossings and extrema. This restriction then allows one
to define the instantaneous frequency for each of the
decomposed IMFs. In other words, an IMF satisfies:
in the whole data set, the number of extrema and the number
of zero-crossings must be either the same or differ at most by one;
(2) at any point, the mean value of the envelope defined by the
local maxima and the envelope defined by the local minima is zero.

23. The first condition is similar to the narrow band requirement for
The Sifting Process
Restriction imposed (Huang et al, 1998):
HHT - Cont’d
The first condition is similar to the narrow band requirement for
a stationary Gaussian process. The second one, however,
modifies the classical global zero-mean requirement to a local
one. It is this very second property that goes with the concept
of the instantaneous frequency that is valid for nonstationary
process and nonlinear signals. From the context of signal
processing, the second property allows us to avoid a
local-averaging time scale altogether.
Invoking the above assumptions and restriction, Huang et al
showed that their empirical mode decomposition (EMD) can
identify the intrinsic oscillatory modes by their characteristic
time scales in the data.

26. Example Problem: Tone plus Chirp Oscillation
(Source: Gabriel.Rilling (at) ens-lyon.fr

27. Mark the maxima

28. Interpolate the maxima by cubic splines
Repeat the minima by cubic splines

29. Obtain the local mean curve, m1
Obtain the residue, r1 = x - m1

30. Iterate on h1 if it violates the assumptions and restrictions

121. Nontrivial Example

124. Converged h1 after 9 iterations

125. Intrinsic Filtering Capability

126. Completeness Demonstration

127. Why the Hilbert Transform?

128. Back to HHT mathematics:
The Hilbert Transform and Instantaneous Frequency
For an arbitrary time series, x(t), its Hilbert transform, y(t), is
defined as
where PV indicates the Cauchy principal value.

129. The Hilbert Spectrum
The Hilbert spectrum: a three-dimensional plot of the amplitude
aj(t) vs time (t) and the instantaneous frequency wj(t), designated as
H(w,t) ,either for each amplitude or the sum of all the amplitudes.

130. Utilizations of the Hilbert Spectrum

131. Comparison of Hilbert Spectrum

132. How good is the instantaneous frequency?
Duffing Equation:

138. Example 2: Period doubling of